Economics 202b: “Bubbles” Handout
In a social-planning or firm intertemporal optimization problem, we often find ourselves
faced with a first-order condition for a co-state variable—that is, the shadow value for the
planner of having an extra unit of resources in place at time t. This first-order condition
often looks something like:
D
 P
Pt  t  Et t 1 
1  r 
1r
In a market-equilibrium problem, we often find ourselves with the same first-order
condition—this time enforced by market arbitrage:
D
 P
Pt  t  Et t 1 
1  r 
1r
And for analytical convenience we often find ourselves assuming (a) that r is constant
over time, and (b) that expected dividend growth is at a constant proportional rate g as
well:
Et(Dt+1)=(1+g)Dt
In the social-planning context we can solve this problem forward:
2
3


1  
1  g 1  g  1 g 
 Pt  4 
Dt 1 

Pt 
 


E
t
3
(1  r ) 
1  r   1 r
1  r  1  r 
 1  s 1 g j
 P

Pt  lim 
Dt  Et  t s 1 s 



s  1  r 
(1 r ) 
j0 1  r

  P
Dt

 lim Et  t s 1s 1 
s
r g
 (1 r) 
And we can then use our second-order condition—the assumption that our social planner
or our decision-making firm is not really stupid—to conclude that:
  P

lim Et  t  s1s1  0
s 
 (1 r ) 
And thus:
Dt
fund
Pt  Pt 
r g
But in the market context, the solution to our arbitrage equation is:
Pt 
Pt = Ptfund + Bt
Where Bt is any term that satisfies:
Et(Bt+1)=(1+r)Bt
And what is there in the market to force Bt=0? Appeals to (far distant) irrationality of the
bubble when it gets really big? Minimal state variables?
Economics 202b: Froot-Obstfeld Handout
Froot and Obstfeld begin with the arbitrage condition and the dividend process:
Pt  e r Et (Dt  Pt 1 )
Dt  e
dt
dt 1    dt   t 1
If we restrict ourselves to solutions that do not depend explicitly on time, the general
solution to this is:
Pt 
Dt
er  e


2
'
 c1 Dt  c2 Dt
2
where  and ’are the roots of:
2 2 / 2    r  0
To see that this is in fact the solution, substitute back in:




Dt
Dt 1

'
r

'

e
E
D

2  c1 Dt  c 2 Dt

2  c1 Dt 1  c2 Dt 1 
t
t
1








er  e 2
er  e 2
Dt

e e
Dt
r

e e
Dt
r
e e
r

2
2
2
'
2
r
 c1 Dt  c 2 Dt' 

e e
Dt
r
2
2


2



Dt e   / 2


t 
'
  t '
 c1 Dt  c 2 Dt  e Dt 

c
D
E
e

c
D
e
2





1
t
t
2
t




r
2


e e
Dt

'
r
 2  2 / 2
r
'   ' 2  2 / 2
 c1 Dt  c 2 Dt 
 c1 Dt  e e
 c2 Dt ' e e
2

e e
r


2
2
2
 c1 Dt  c2 Dt'



Economics 202b: “Noise” Handout
Milton Friedman has been perhaps the most powerful advocate of the position that asset
market institutions—limited job tenure of portfolio managers, short horizons of investors,
people who buy and sell for not-very-good reasons—don’t matter. The argument that
they don’t matter has two parts: (a) recursion and (b) selection.
Recursion—A linked series of investors each in the market for one period should
perform the same calculations as one very long-term investor with the same rate
of time discount.
Selection—People who don’t optimize buy high and sell low: they lose money,
and exit from the market.
But recursion only works if there is no “noise” in the market. If there is noise, risk-averse
stabilizing speculators will want to take limited positions only. And people who don’t
optimize lose utility—not necessarily wealth.
In the model investors “live” for two periods, consume in the second period only, and
maximize:
U = -e, which with normal returns is equivalent to max E() – 2

Risky asset—available in unit supply—pays a constant dividend r and sells for a price pt;
safe asset pays a constant dividend r and sells for a price of 1 at all times. “Noise traders”
( of them) misperceive the expected price of the risky asset next period by a normally
distributed (but not mean zero) random variable .
Rational investors (present in the model in measure 1-) choose to hold an amount it of
the risky asset to maximize:
E(U) = c0 + it [r +Et pt+1 – pt(1+r)] – (t2p(t+1))(it)2
Noise traders choose to hold an amount nt of the risky asset to maximize:
E(U) = c0 + nt [r +Et pt+1 – pt(1+r)] – (t2p(t+1))(nt)2 + nt t
Agents’ holdings of the risky asset are:
r  Et pt 1  (1  r) pt
it 
2 (t  2p t 1 )
it 
r  Et pt 1  (1  r) pt  t
2 (t  2p t 1 )
Setting total holdings equal to supply gives us:
r  Et pt 1  2 t 2pt 1  t
pt 
1 r
And looking for a stationary equilibrium gives us:
2 2
 (t   *)  * 2  
pt  1 


1 r
r
r(1  r)2
The expected difference in returns between noise traders and sophisticated investors is:
E(Rni )   * 
(1 r )2  *2 (1  r) 2  2
2 
2
And in an extension of the model in which there is fundamental risk:
(1 r)  * (1  r)  
2 (1 r)2  2
2 2 
2
E(Rni )   * 
2
2

2